The Great Pyramid at Giza (Giza 1) is a
gem in pyramid building, and receives virtually all of the attention. But it
was not built in isolation. It was part of a construction project that
included three other large pyramids. One of those is the Giza companion (Giza
2). The other two are located at Daschur, usually denominated the Flat and the
Bent. Together, those four constitute more than 70% of the total Egyptian
pyramid volume.

The step pyramid at Saqqara, and the
Meidum pyramid appear to be evolutionary projects approaching the size of the
large four, but all other pyramids are puny in comparison.

Examination of the size and the slope of
the exteriors of the four great pyramids show them related to one another
mathematically.

I shall review the technical data of each
in turn, in the sequence of the Bent, the Flat, and Giza 2, while reserving
most attention for Giza 1. I shall then show how they are related.

My sources are:

1. W. M. Flinders Petrie, The Pyramids
and Temples of Gizeh, Field and Tuer, London, 1883, First Edition.

2. J. H. Cole, Determination of the
Exact Size and Orientation of the Great Pyramid of Egypt, Survey of Egypt,
Paper #39, Government Press, Cairo, 1925.

Maragioglio and Rinaldi summarized the
data collected by most other previous workers. Some of this data was reported
from sources not cited above. Also, their work was not always the most
accurate. So we must be careful in accepting their data. Hereafter I refer to
this work only as Maragioglio.

The Pyramids at Daschur

The North pyramid at Daschur is the Flat;
the South pyramid is the Bent. They are a companion pair, the same as the
companion pair at Giza.

North Pyramid at Daschur

Maragioglio twice reported Perring but
differently in the two reports.

Maragioglio reporting Perring (#1)

Base 219.42 m (418.74 cubits)

Height 104.49 m (199.41 cubits)

Slope 43o 36' 11"

Maragioglio reporting Perring (#2)

Base Length 219.28 m (418.47 cubits)

Height 104.42 m (199.27 cubits)

Slope 43o 36' 11'

Maragioglio reporting Reisner

Base 218.5 m (416.98 cubits)

Height 104.4 m (199.23 cubits)

Slope 43o 31' 11"

The slope of a 20-21-29 integral
Pythagorean triangle is 43o 36' 10".

The Flat pyramid at Daschur was designed as a
20-21-29 integral Pythagorean triangle.

The height of the pyramid was intended to be 200
cubits.

The base length then was determined.

Ideally, the base would be 2 X 209.81 = 419.63
cubits.

This compares to the measured values of
418.74 and 416.98 cubits.

South Pyramid at Daschur

This pyramid is composed of a frustum,
topped by a true pyramid shape.

Maragioglio reporting Petrie

Base 188.6 m (359.92 cubits)

Bottom slope 54o 31' 13"

Upper slope 43o 21'

Height from base to bend 49.07 m (93.64
cubits)

Height from bend to top 56.00 m (106.87
cubits)

Total height (200.51 cubits)

Unfortunately, we do not have comparison
measurements from other investigators.

Maragioglio personally observed great
cracking of the stones comprising the structure. There was settling, not due
so much to earthquakes, but to compression of the ground. Therefore, the
observed dimensions and slopes may not exactly reflect the original
construction.

Given these precautions we can draw some
conclusions.

The height of the Bent was designed the same as
the Flat at 200 cubits.

The base measurement suggests it was designed at
360 cubits.

In order to show the design parameters I
sketched an outline of the pyramid, compared against the Flat. The figure
shows the design ratios of the various segments.

If the pyramid was designed at 360 base,
it was shortened from the base of the Flat by 119.63 cubits.

To help analysis I divided one half of
the base of the Flat pyramid into 63 parts. This made each part slightly short
of 3 1/3 cubits.

Assuming 3 1/3 cubits, the Bent half-base
length would be 54 X 3 1/3 = 179.84 cubits. The full length would be 359.68,
or short of 360 by .32 cubits, equal to 6.6 inches.

Or, doing it the other way, assuming that
the Bent base is the criteria for dividing the length into integral portions
of 3 1/3 cubits, the length of the Flat base would be 60 cubits longer than
the Bent, or 420 cubits. This differs from the ideal of 419.63 by 0.39 cubits,
or eight inches.

If the Flat one-half base were divided 63
parts for analysis, and maintaining the ratio of 20/21, the height would be
divided into 60 parts. This would again give 3 1/3 cubits per part.

From this analysis the height of the bent
section would be (28/60 X 200) = 93 1/3 cubits. The measured value is 93.64.
The difference is 0.3 cubits from the assumed design ideal.

The height of the top portion would be
(32/60 X 200) = 106 2/3 cubits. The measured value is 106.87 cubits. The
difference is 0.27 cubits.

These two errors add to 0.57 cubits.

Clearly, the ravages of time and the
errors of measurement prevent us from determining more precisely how the
designers proceeded with their construction.

This line of analysis leads us to believe
that the slope of the frustum was 7/5, or an angle of 54o 28'. This
compares with a measured value of 54o 31'.

The slope of the pyramid top would then
be 0.94 for an angle of 43o 15'. This compares to 43o
21' measured. (0.94 is equal to a slope of 16/17)

We can see from the diagram that the bent
portion protrudes slightly above the slope of the Flat pyramid, if we assume
these design criteria. The calculated amount is about 1.6 cubits, 33 inches,
or about one yard.

Ideally, we would desire the bend of the
Bent pyramid to fall directly on that slope line. The interested reader can
perform the calculations to determine how much the base and height values
should change to achieve that result.

The Bent pyramid outline apparently was intended
to fall within the outline of the Flat,

while constricted by the suggested design
criteria.

These design criteria would imply that
the pyramid builders used simple arithmetic ratios to achieve their measures.

Giza 2 appears to be designed as as 3-4-5
integral Pythagorean triangle.

Note that the base length of Giza 2 of
410.80 cubits is less than the Flat base length of 419.63 cubits by about nine
cubits. I shall return to further discussion.

Giza 1

From Petrie=s
measurements:

North side = 9069.4 inches = 439.62
cubits

East side = 9067.7 inches = 439.54 cubits

South side = 9069.5 inches = 439.63
cubits

West side = 9068.6 inches = 439. 58
cubits

Measured slope = 51o 50'
40" (This was a weighted mean up the north face.)

Petrie noted that Athe
South face should not be included with the North, in taking the mean, as we
have no guarantee that the Pyramid was equiangular, and vertical in its axis.@

Petrie gives calculated height at 5,776
+/- 6 inches. This would be 279.98 +/- 0.29 cubits, for an ideal of 280
cubits.

From Coles measurements:

North side = 230.253 meters = 439.41
cubits

East side = 230.341 meters = 439.58
cubits

South side = 230.454 meters = 439.80
cubits

West side = 230.357 meters = 439.61
cubits

Thus we can see that Cole differed from
Petrie on the average around the four sides by no more than +0.03 cubits.
Their differences overall appear to be no more than error of measurement.

The difficulty in determining the design
criteria for the Great Pyramid is its proximity to three different solutions.

#1: The height of the pyramid represents
the radius of a circle, while the base perimeter represents the circumference
of the same circle.

It is expressed mathematically as

h = 2 Pi X 4s

#2: The square of the height is equal to
the face area of one side.

This is the famous Golden Ratio, or
Golden Section, also known as the Divine Ratio.

This ratio was used as a design guide in
many ancient structures, has been discussed extensively in mathematical and
architectural literature, and is well known in the field of art.

It is expressed mathematically as

a/b = b/(a + b)

The ratio is mathematically unique.
Numerical solutions from the above equation are 0.618034 and

-.618034.

I shall return to further discussion of
this unique ratio.

#3: The area of a circle is to the square
of its diameter as 11 to 14.

This ratio was given by Archimedes (c.
287-212 BC) in his treatise, Measurements of a Circle, Proposition 2.

This is the famous Pi approximation of 3
1/7 = 22/7.

We can now calculate the pyramid
dimensions subject to these three different criteria.

From #1:

If the radius of the hypothetical circle
is the height at 280 cubits, then the circumference is 2 Pi X h = 1,759.3
cubits. The sum of the Petrie values is 1758.37.

Certainly this hypothesis is correct,
within measurement error. The difference is only 0.93 cubits.

From #2:

Solutions based on the Golden Ratio are
found as follows:

Let the ratio of one-half the base
length, x, to the height, y, be equal to the ratio of the height, y, to the
apothem, a.

x / y = y / a

But from the Pythagorean theorem we know
that a2 = (x2 + y2).

In order to make this tractable we must
square both sides.

x2 / y2 = y2 /
(x2 + y2)

This equation is the same form as the
Golden Ratio, except that it is expressed in terms of the squares of the
elements.

Thus we find that mathematically x2
/ y2 = 0.618034.

We obtain 0.616 from the ratio of the
measured values of 219.82 / 2802.

The slope of the ideal Golden Ratio is x
/ y, or the inverse of the square root of 0.618034 = 0.78615. The inverse is
1.2720.

The slope from the measured values is
280/219.8 = 1.274.

Hence we could now conclude that the
Great Pyramid was designed around the Golden Ratio.

From #3:

The slope of 11/14 is 0.7857. The inverse
is 1.2727.

Compare this with the measured values,
and also with the slope determined from the Golden Ratio.

Hence we could conclude that the simple
Archimedes ratio of 11/14 was the criteria for the construction of the Great
Pyramid.

This would agree with our earlier
assessment that the builders used simple arithmetic ratios for their design.

The uncanny part of these criteria is
that the values of three important hypotheses all fall within the construction
tolerances of the pyramid.

Someone certainly knew his mathematics.

The ideal angles for these three are as
follows:

#1 51o 51.2'

#2 51o 49.6'

#3 51o 50.6'

Petrie gave 51o 50' 40"
(51o 50.6') for the North face weighted mean.

Hence Petrie=s
weighted mean falls in the mid-range of the theoretical ideals, and almost
exactly on the 11/14 ratio.

The ideal slopes for these three are:

#1 1.2732

#2 1.2720

#3 1.2728

Petrie=s
measured slope is 1.274.

Petrie noted differences among the four
sides of the pyramid. A concavity exists on each of the sides, and makes the
exact determination of construction intent impossible.

Hence we shall never be able to conclude one way
or the other which of these three criteria were used to design the Great
Pyramid.

Putting Pieces Together

Petrie was aware of the integral
Pythagorean triangles in the pyramids. He said that the use of ratios:

A.
. . seems to suggest that the square of the hypothenuse being equal to the
squares of the two sides may have been known; particularly as we shall see
that the use of squared quantities is strongly indicated in the Great
Pyramid.@But many other elements tie these Great
Pyramids to one another.

All four integral heights seem to be the
starting point for calculating the base lengths. These are 200 cubits for the
two pyramids at Daschur, 275 cubits for Giza 2, and 280 cubits for Giza 1.

Now a curious question raises itself. If
Giza 2 were increased in height to that of its companion, 280 cubits from 275,
as we see for the two pyramids at Daschur, the length of the base would
increase from 410.8 to 421.2 cubits.

The base length of the Flat is 418.5
cubits.

These two values differ by only 3 cubits.

If we were to sketch the outlines of the
four pyramids to scale for this idealized scheme we would discover that the
two Giza pyramids would touch at the top, the two Daschur pyramids would touch
at the top, the Bent falls within the outline of the Flat, and the Flat and
Giza 2 very nearly coincide at the base.

Refer to the two sketches, one showing
the designs as they were built, and the other according to this idealized
scheme.

Outline of the Four Great Pyramids - As Built

Outline of the Four Great Pyramids - Ideal

Given the refinement of the great pyramid
designs, and their clear connection to one another, why did the designer not
make the Giza 2 height equal to Giza 1?

The cubit dimensions for Giza 2 of x, y,
and a are 3/2 X 137, 4/2 X 137, and 5/2 X 137. 137 is a prime number. Prime
numbers are used elsewhere in pyramid construction. Thus it is possible that
the designer intended to make this display of mathematical knowledge, rather
than simply equating the two Giza pyramid heights.

Such design ratios are in the other
pyramids.

The Flat uses multiple of 10 to obtain
the cubit measures: 20 X 10 for the height y, 21 X 10 for the base x, and 29 X
10 for the apothem, a.

If the Great Pyramid was designed
according to the Archimedes ratio it uses multiples of 20 to obtain the cubit
measures: 14 X 20 for the height y and 11 X 20 for the base x. The ratio of
14/11 does not provide an integral Pythagorean solution for the apothem, a.

Note that Giza 2 with a height of 280
cubits would have had multiples of 70 to obtain 3, 4, and 5 X 70, instead of
multiples of a prime number.

The Bent pyramid uses multiples of 40/3
and 20/3 to obtain the simple trigonometric ratios.

The ratio of the height of the frustum to
the base of the frustum is 93 1/3 to 66 2/3, or 7/5. The multiplication fact
to obtain the cubit lengths is 40/3. Stated otherwise:

(93 1/3)/7 = (66 2/3)/5 = 40/3

The top of the Bent has a multiplication
factor of 20/3:

(106 2/3)/16 = (113 1/3)/17 = 20/3.

The ratio of the two multiplication
factors is 2:1.

These factors become more interesting
when it is recognized that the frustum of the Bent contains numbers related to
Giza 1 and the Flat.

The height of the frustum is 93 1/3 =
280/3 cubits. The width of the frustum is 66 2/3 = 200/3 cubits. Therefore the
height of the Bent frustum is exactly 1/3 of the height of Giza 1, while the
base of the frustum is exactly 1/3 the height of the Flat.

Or, to put it another way, the ratio of
the height of Giza 1 to the height of the Flat is 7/5, and this is the slope
of the Bent frustum.

Other factors should be noted:

Decimal multiples are used in Giza 1 (20)
and the Flat (10). One is at Giza; one is at Daschur.

Fractional multiples (1/3) are used in
the Bent. A primary number is used in Giza 2. One is at Giza; one is at
Daschur.

Giza 2 and the Flat use integral
Pythagorean relationships, 3-4-5 and 2B21-29.
One is at Giza; one is at Daschur.

Giza 1 and the Bent do not use integral
Pythagorean relationships. One is at Giza; one is at Daschur.

The decimal multiples and the integral
Pythagorean numbers are not in the same structures at Giza and Daschur.

These design elements can be seen more
clearly in the following table:

Design
Multiples

Decimal

Fractional

Relationships

Integral

Flat

Giza 2

Non-integral

Giza 1

Bent

This arrangement of elements can hardly
be accidental, and shows intent by the designer to tie all four pyramids
together in one grand architectural scheme.

The silliness of previous speculations
can be noted from this remark by I. E. S. Edwards, who once held a wide
reputation for his knowledge of the pyramids:

AThe
temptation to regard the true Pyramid as a material representation of the
sun=s
rays and consequently as a means whereby the dead King could ascend to
heaven seems irresistible.@Such primitive notions would have been a
great disappointment to the greatest architectural genius of all historic
time.